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Ciliberto, Ciro; Zaidenberg, M.

Scrolls and hyperbolicity. (English) Zbl 1270.14027

Int. J. Math. 24, No. 4, Article ID 1350026, 25 p. (2013).
Several questions concerning very general hypersurfaces of degree \(d\) in the projective space \(\mathbb{P}^n\) are of interest, for instance the following are discussed in the paper under review: (1) to know what is the lowest geometric genus of a reduced and irreducibe curve lying on these hypersurfaces; (2) to bound the geometric genus (or other numerical invariants) of higher-dimensional subvarieties of them; (3) to know if they do not admit a non constant morphism from an abelian variety, algebraic hyperbolicity; (4) to know if they do not admit a non-constant entire curve, Kobayashi hyperbolicity; (5) to know if there exists a real number \(\epsilon>0\) such that any algebraic curve on the hypersurface verifies \(2g(C)-2\geq \epsilon \deg C\), Demailly algebraic hyperbolicity.
In this paper, the authors use a degeneration method of the general hypersurface to a special one (mainly scrolls) to deal with this kind of questions. In particular they provide a proof of the non-existence of low genera curves on general surfaces of degree \(\geq 5\) of the three-dimensional projective space (see Section 2). They also provide a bound of the geometric genera of surfaces on general threefolds in \(\mathbb{P}^4\) (see Section 3). Finally, they prove the existence of hyperbolic surfaces (resp. threefolds) of any degree \(d \geq 6\) (resp. \(d\geq 12\)), being unknown the case \(d=7\), in \(\mathbb{P}^3\) (resp. \(\mathbb{P}^4\)), showing consequently their Demailly algebraic hyperbolictiy (see Theorem 4.6 in Section 4).
Reviewer: Roberto Munoz (Madrid)

Cited in 4 Documents

MSC:

14N25 Varieties of low degree
14J70 Hypersurfaces and algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds

Keywords:

Kobayashi hyperbolicity; algebraic hyperbolicity; projective hypersurface; geometric genus; degeneration methods
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References:

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